The Nachbaur Diode Limiter, Part 2: Selecting the resistors

This post is part two in a series of three.
Part one discussed the basics.
Part three discusses some other variations on the circuit.

It's not difficult to determine the resistances to use in the Nachbaur diode limiter circuit, starting with an arbitrary set of input voltages for each knee point in the curve, as long as the constraints discussed in part one are met.  The following spreadsheet (on Google Sheets) will perform the calculation: Nachbaur Limiter Calculator (voltage input)

If you select output voltages that don't meet the constraints mentioned in part one, you will get errors or odd outputs like negative-valued resistors.  You can save a copy of this sheet to your own Google Drive to edit.

Instead of using some arbitrary voltages, I wanted to fit a smooth curve with a fairly simple definition to the transfer function, and use that to set the voltages.  Mostly, I wanted to select:

1. The input voltage corresponding to the last knee in the curve.
2. The number of diodes in the ladder.
3. The final gain (or limiting ratio).

A few notes on the math that comes out of this:

• It's much easier to fit a curve to the inverse of the transfer function, since the diode voltage drops constrain the output voltage to known values.  This avoids having to derive inverse functions to perform the resistor selection.  Thus, this is the approach I took.
• You can fit a parabola, V_in(v) = av² + bv + c, or a cubic function V_in(v) = av³ + bv² + cv + d, several different ways, but for a parabola, the final gain cannot be selected; for either polynomial, the curve meeting all the requirements is not guaranteed to be monotonic.  If you consider the first diode's part of the curve to be linear, you also have to constrain the slope of the polynomial at that point.
• You can fit a simple power law curve V_in(v) = a (bv)^p to the points.  In order for the ratio to increase, it requires p > 1.  You also can't constrain the slope at the end.    (If you choose the slope at the end, you can't constrain the input voltage at the final knee).

I also tried some experiments with combinations of linear and exponential/hyperbolic functions, but the curves tended to be too "hard", with impractically large resistance values at one end of the ladder,
and very small on the other.

The curve I ended up choosing as the "best" is a combination of a linear term and a power law scaled to fit.  The power law is then easily shaped by exponent (to get the right final ratio).  It's also guaranteed to be positive, monotonically increasing, and with a first derivative (limiter ratio) increasing from zero (as long as the exponent is greater than one), which is important for its smoothness when connected piecewise to V_in(v) = v (which describes the behavior before the first diode turns on).

The following spreadsheet will perform the calculation:  Nachbaur Limiter Calculator (power+linear)

Fitting a continuous function was not really necessary though, since we're approximating the diodes' behavior (diodes turn fully on at once, and have a fixed voltage drop) and we're only calculating the inverse transfer function at integer multiples of a diode forward voltage drop anyway.  I came up with this discrete option: If d represents a diode voltage drop, and p represents the final limiter ratio, we can calculate the increase in required input voltage at step N as d p^i/N. The input voltage after step N then is the sum:

...which doesn't seem so bad, until you try to solve for p, so that you can choose the final knee voltage.  In that case, the solution requires the Lambert W function, which spreadsheets don't seem to have and cannot be expressed in terms of elementary functions.  I guess you could just tinker with it until it works, or use your spreadsheet's goal seek function.

This spreadsheet will perform the calculation: Nachbaur Limiter Calculator (exponential ratio increase)

Interestingly, it turns out that the combination of resistors generated by this version resembles Fred's original resistor choices most closely. Just set your final limiting ratio to 9 to match his.  Also, when we get to Part three, we'll see that there's a way to tailor this version to fit the required voltages and ratio, while solving some other practical issues at the same time.

Although all the mathematics were fun, it turns out that for my planned application, there is a practical problem to using this circuit with any combination of resistors: An implementation with many diode steps requires a higher input voltage swing than I can manage. For the last limiter step to do its job, the input signal needs to go well beyond the highest knee voltage.  Some other tweaks are necessary.   But, if you're doing a tube circuit that can manage the voltage swing, you can still use it as-is.

Next post: Other variations.